Coordinated Control of Multiple UAVs : Theory and Flight Experiment.

HAL Id: hal-01061122

https://hal-onera.archives-ouvertes.fr/hal-01061122

Submitted on 5 Sep 2014

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Coordinated Control of Multiple UAVs : Theory and Flight Experiment.

Y. Watanabe

To cite this version:

Y. Watanabe. Coordinated Control of Multiple UAVs : Theory and Flight Experiment.. AIAA Guid-

ance Navigation & Control (GNC) Conference, Aug 2013, BOSTON, United States. �hal-01061122�

Coordinated Control of Multiple UAVs : Theory and Flight Experiment

Yoko Watanabe ^{∗ ∗}

ONERA - The French Aerospace Laboratory, Toulouse 31055, France

This paper proposes a nonlinear control law to realize a coordinated flight of multiple UAVs and evaluates its performance through flight experiments of small fixed-wing UAV platforms. Assuming all-to-all communication, a decentralized coordination control system is designed based on a virtual leader approach. The proposed control design uses a po- tential function defined on a phase distribution of multi agents. Two advantages of this coordination controller are; i) it can be applied to make different coordination configura- tions, ii) it is applicable to any number of UAVs, and so it can easily treat an event of addition/deletion of UAV units in a coordination team. The proposed coordination control law is proven to be locally asymptotically stable by using Lyapunov indirect method, and its large domain of attraction is observed in simulation. Furthermore, the controller is im- plemented onboard ONERA fixed-wing UAV platforms and tested with a mission scenario which includes four different coordination configurations.

I. Introduction

Coordinated operation of multiple unmanned aerial vehicles (UAVs) has aroused many researchers’ in- terest in recent years, due to its great potential for increases in overall mission performance and robustness without augmenting a capacity of each UAV unit. The followings are some examples showing advantages of multi-UAV operation over mono-UAV one. For a surveillance mission, even though each vehicle can carry a limited number/type of sensors, data fusion of several of them distributed over an operation site in a desired configuration can provide further information and a wider sensing coverage. Communication relay can be established between vehicles to extend a range of datalink connection with a ground control station.

System robustness can be improved by making redundancy so that a functional failure of one UAV among a team does not necessarily lead a mission failure. There is also an aerodynamic benefit to reduce a power consumption of each aircraft when flying in formation

¹

.

Various strategies for multi-agent coordination have been investigated in robotics community since many years. Especially, formation control of multiple robots, aircrafts, underwater vehicles or satellites has been intensively studied

²

. There are two objectives in a coordination control problem. One is to make each agent maintain a desired geometric configuration with respect to others. The other is to make an entire team achieve a mission objective such as reference trajectory tracking. Two main approaches for such coordination control are i) leader-follower approach and ii) virtual structure approach. In the leader-follower approach, an agent designated as a leader takes charge of a mission objective and the other agents designated as followers maintain a desired configuration with respect to the leader. A big advantage of using this approach lies in its facility of implementation. As shown in Figure 1, the leader-follower coordination control architecture does not have any feedback loop from followers to the leader (C and D blocks in the figure denote a local controller and an agent dynamics respectively, for each agent.). This is why most of the existing work of successful UAV formation flight adopt this approach. Bayraktar et al.

³

and Dong et al.

⁴

realized a formation flight between two fixed-wing airplanes and between two helicopters respectively, by updating a follower’s flight plan using a current leader’s position in real-time on ground control station. In the work of Gu et al.

⁵

, three airplanes flew in formation by transmitting a leader’s position directly to the two followers by radio communication. Johnson et al.

⁶

developed a vision-based system onboard a follower aircraft to localize a

∗

Research Engineer, Systems Control and Flight Dynamics Department, [email protected]

Figure 1: Leader-Follower Coordination Control Architecture

Figure 2: Virtual Structure Coordination Con- trol Architecture

Figure 3: Virtual Leader-Follower Coordination Control Architecture

leader and achieved a two-UAV closed-loop formation flight without any communication between the UAVs.

However, this leader-follower approach has a weakness in its robustness. As followers’ motion is determined relative to their leader’s, an error in the leader’s maneuver is fully propagated to all the followers and both the mission and coordination objectives fail once losing the leader’s performance.

In order to overcome this robustness issue, the virtual structure approach was introduced by Tan and Lewis

⁷

. Virtual structure (VS) is defined as a collection of agents, which maintains a desired geometric configuration. This approach consists of the following three steps; 1) VS is aligned with the current agent positions; 2) mission control is applied to VS to obtain a desired trajectory for each agent; and 3) each agent tracks its own desired trajectory. The mission and coordination objectives are both managed by a centralized VS control system with a feedback of every agents’ states, as shown in Figure 2. However, it can be easily decentralized by duplicating the VS control system on each agent. Due to its capability to maintain a highly precise formation, many propose a spacecraft or satellite formation control law based on the VS approach

⁸

. For multi-UAV control application, only few work applies the VS approach. This is because of a high computational complexity of the VS position and orientation fitting from the current agents’ states.

Although Low and Ng

⁹

suggest using a dynamic virtual structure along a maneuver-constrained trajectory for UAV formation control, it omits the first VS alignment step by assuming a given formation trajectory.

So to be exact, this work does not apply the original VS approach but rather the virtual leader approach.

The virtual leader (VL) approach introduced by Leonard and Fiorelli

¹⁰

is a combination of the two

approaches discussed above, although it is often categorized as a family of the VS approach. Virtual leader

state is determined by all the agents’ current states, as a virtual structure in the VS approach. Then

each agent tries to maintain a desired geometric configuration relative to the VL, as in the leader-follower

approach. Figure 3 depicts a control architecture of the VL approach. Commonly, the VL position is taken

at a center of mass of a team of all the agents. Hence the VL determination does not require a complex

optimization process as required in the VS alignment step. Moreover, the VL approach remains more robust than the leader-follower approach in a sense that a failure of one agent will be only partially propagated to the others via the VL position. Since the VL approaches overcome the drawbacks of both the leader-follower and the VS approaches, some have proposed its application to UAV formation flight

¹¹

, but not yet many examples of actual flight result using this approach can be found.

In many literatures

^2,8

, a behavioral approach is listed as the third approach of coordination control. This approach derives a control strategy as a weighted average of several different task controls such as trajectory tracking, formation keeping and obstacle avoidance

¹²

. The coordination/formation keeping task control is essential in the behavioral approach, and it can be designed based on the two approaches mentioned above.

For this reason, we prefer to distinguish this approach from the other two.

In this paper, a coordination control law for multiple UAVs is designed based on the decentralized VL approach. Inspired by a notable work of Leonard et al.

¹³

and Sepulchre et al.

¹⁴

on multi-agent collective motion control, we propose a controller with which each agent achieves;

• a stable circling motion at constant radius and speed around a VL position,

• a splay or synchronized phase distribution with the other agents on the circle orbit, and

• a global mission objective.

A global stability and local asymptotic stability of the desired coordination state using the proposed controller are theoretically proven by using Lyapunov direct and indirect methods. An interesting aspect of this controller is that it can be applied to make different types of coordination flight by choosing different configuration settings. For example, a splay or synchronized circling is realized by taking a fixed beacon as a virtual leader with a non-zero circling speed. Trajectory tracking in formation can be attained by taking a reference trajectory as a virtual leader with zero circling speed. We can also take a virtual leader in a conventional way, at a center of mass of the team, and use this coordination controller in combination with a VL mission controller. Another feature is that the proposed controller is applicable to any number of agents in the team. A number of agents is used as a parameter in the controller, and there is no limitation on it. This feature allows the coordination control system to handle easily an event of addition/deletion of an agent to/from the team even during operation.

The coordination controller proposed in this paper is implemented onboard three of the ONERA fixed- wing UAV platforms, shown in Figure 4, developed based on Multiplex TwinStarII R/C airplane. A mission scenario is defined by a sequence of four different coordination phases; 1) coordinated circling; 2) synchronized circling; 3) circling in formation; and 4) waypoint tracking in formation. As mentioned above, all of these phases can be realized by the same coordination control algorithm. Transitions between the phases are made autonomously by changing the configuration settings of the controller when certain conditions are satisfied.

In most of the related experimental work of UAV formation flight, a transition from non-coordinated to coordinated flight phases is performed deliberately. It is often the case that a safety pilot manually places the follower UAV at a desired relative position to the leader before activating an automatic formation control

^4,5

. Unlike those work, we have achieved a completely automatic coordination flight of multiple airplane UAVs without any manual adjustment. The flight test results are shown as a highlight of this paper.

II. Multi-Agent Coordination Control

This section provides a coordination control design and its proof of stability. As stated in Section I,

this paper proposes a coordination controller based on the decentralized virtual leader approach. Assuming

all-to-all communication, each agent determines independently the virtual leader state and coordination

control inputs by using available information on other agents’ states. Before presenting the control design,

an (M, N)-pattern phase distribution defined in Leonard et al.

¹³

and in Sepulchre et al.

¹⁴

is introduced.

Figure 4: The ONERA Fixed-Wing UAV Platforms

A. (M, N)-pattern phase distribution

Let θ ∈ T ^N be a vector of N phase (angle) states, where T = (0, 2π]. Define the m-th moment of the phase distribution θ as follows.

p _m (θ) = 1 mN

X N

k=1

· cos mθ k

sin mθ k

¸

(1) Then its potential can be defined by

U m (θ) = N

2 kp _m k

²

= 1 2m

²

N

X N

k=1

X N

j=1

cos m(θ k − θ j ) (2)

Theorem II.1 The potential U m reaches its unique minimum of U m = 0 when p _m = 0, and its unique maximum of U m = N/2m

²

when the phase difference between any two phases is an integer multiple of 2π/m.

Proof See Theorem 5 in Sepulchre et al.

¹⁴

Let 1 ≤ M ≤ N be a divisor of N . An (M, N)-pattern is a symmetric arrangement of N phases consisting of M clusters uniformly spaced around the unit circle, each with N/M synchronized phases. Figure 5 shows an example of the (M, N )-pattern phase distributions when N = 12. Then the (M, N )-potential function, denoted by U ^M,N , is defined by

U ^M,N (θ) = X M

m=1

K m U m (θ) (3)

Figure 5: (M, N )-Pattern Phase Distributions (Case N = 12) : Size of a circle corresponds to a number of

agents (= N/M) in a synchronized phase.

Theorem II.2 θ ∈ T ^N is an (M, N )-pattern if and only if it is a global minimum of the potential function U ^M,N (θ) with K m > 0 for m = 1, 2, ..., M − 1 and K M < 0.

Proof See Theorem 6 in Sepulchre et al.

¹⁴

Two special cases of (M, N )-pattern phase distribution are; i) synchronized state when M = 1 and ii) splay state when M = N. In this paper, those two cases are applied to a coordination/formation control of multi-agents.

B. Coordination Control Design with Given Virtual Leader Trajectory

Let X _k (k = 1, 2, . . . , N ) and X _vl be a 2D position of the k-th agent and a virtual leader in an inertial reference frame. Define a relative position and velocity of the k-th agent to the virtual leader by

r k = X k − X vl = r k

· cos θ _k sin θ k

¸

(4)

˙

r k = X ˙ k − X ˙ vl = ˙ r k

· cos θ k

sin θ k

¸ + r k θ ˙ k

· − sin θ k

cos θ k

¸

(5) where r k ≥ 0 is distance and θ k ∈ T is phase. Define the following sets.

P = {k ∈ {1, 2, . . . , N } | r k = 0, r ˙ k = 0}

Q = {k ∈ {1, 2, . . . , N } | r k = 0, r ˙ k 6= 0}

R = {k ∈ {1, 2, . . . , N } | r k 6= 0}

The polar coordinate system states for each set can be determined from r k and ˙ r k as shown in Table 1.

Table 1: The Polar Coordinate System States

r k r ˙ k θ k θ ˙ k

k ∈ P 0 0 unspecified unspecified

k ∈ Q 0 k r ˙ k k tan

⁻¹

³ r

˙_kY

˙

r

_kX

´

unspecified k ∈ R kr k k

· cos θ _k sin θ k

¸ _T

˙

r k tan

⁻¹

³ _r

r

kY_kX

´ ·

− sin θ _k cos θ k

¸ _T r

˙k

r

_k

Theorem II.3 Given a virtual leader trajectory (position, velocity and acceleration). With the following control law (6-8) for each agent, every agent’s trajectory converges to a circle orbit around the virtual leader trajectory with a radius r o (> 0), an angular velocity ω o and a phase configuration in a critical set of U ^M,N (θ).

X ¨ k = u _rk

· cos θ k

sin θ k

¸ + u _θk

· − sin θ k

cos θ k

¸

+ ¨ X vl (6)

where

u _rk =

 



K r r

0

, k ∈ P

K r r

0

− K r

˙

r ˙ k , k ∈ Q

−K r (r k − r o ) − K r

˙

r ˙ k − K rθ r k ( ˙ θ k − ω o )

²

− r k θ ˙ _k

²

, k ∈ R

(7)

u _θk =

 



 

0, k ∈ P

2 ˙ r k ω

0

, k ∈ Q

− _r

¹

k

³

K θ ∂U

^M,N(

θ

⁾

∂θ

k

+ K θ

˙

( ˙ θ k − ω o )

´

− (1 − K rθ ) ˙ r k ( ˙ θ k − ω o ) + 2 ˙ r k θ ˙ k , k ∈ R

(8)

with positive gains, and θ k for k ∈ P is chosen so that ∂U ^M,N (θ)/∂θ k = 0. Furthermore, the splay and

synchronized states are locally asymptotically stable when M = N and M = 1, respectively.

Proof With the controller (6-8), the relative distance dynamics of the k-th agent become

¨ r k =

 



K r r

0

, k ∈ P

K r r

0

− K r

˙

r ˙ k , k ∈ Q

−K r (r k − r o ) − K r

˙

r ˙ k − K rθ r k ( ˙ θ k − ω o )

²

, k ∈ R The relative phase dynamics are not specified for k ∈ P, ˙ θ k = ω

0

for k ∈ Q, and

θ ¨ k = − 1 r

²

_k

µ

K θ ∂U ^M,N (θ)

∂θ k + K θ

˙

( ˙ θ k − ω o )

¶

− (1 − K rθ ) r ˙ k

r k ( ˙ θ k − ω o ) for k ∈ R. Define the system state by

x =



  r

˙ r θ ˙ θ



 

where r = [ r

1

· · · r N ] ^T and θ = [ θ

1

· · · θ N ] ^T , and its domain by D = (R

^+N

× R ^N × T ^N × R ^N ). The dynamic system of the state x has the following limit cycle.

½ r k = r

0

, r ˙ k = 0, θ ˙ k = ω

0

∀ k ∈ {1, 2, ..., N }

θ of an (M, N )-pattern phase distribution (9)

Define a potential function as below.

V ^M,N (x) = 1 2K θ

X N

k=1

n

K r (r k − r o )

²

+ ˙ r _k

²

+ r

²

_k ( ˙ θ k − ω o )

²

o

+ U ^M,N (θ) (10)

where U ^M,N (θ) defined in (3). This function is lower-bounded by K M N/2M

²

, which is attained uniquely at the limit cycle (9). A time-derivative of this potential function V ^M,N (x) is derived as follows.

V ˙ ^M,N (x) = X N

k=1

θ ˙ k ∂U ^M,N (θ)

∂θ k

+ K r

K θ

X N

k=1

(r k − r o ) ˙ r k + 1 K θ

X N

k=1

³

˙

r k (¨ r k + r k ( ˙ θ k − ω o )

²

) + r

²

_k ( ˙ θ k − ω o )¨ θ k

´

= − K r

˙

K θ

X

k /

∈P

˙ r

²

_k − K θ

˙

K θ

X

k∈R

( ˙ θ k − ω o )

²

+ ω

0

X

k /

∈P

∂U ^M,N (θ)

∂θ k

+ X

k∈P

θ ˙ k ∂U ^M,N (θ)

∂θ k

The last two terms vanish as

∂U ^M,N (θ)

∂θ k = 0 for ∀ k ∈ P , X

k /

∈P

∂U ^M,N (θ)

∂θ k = X N

k=1

∂U ^M,N (θ)

∂θ k = 0 Hence, we have

V ˙ ^M,N (x) = − K r

˙

K θ

X

k /

∈P

˙ r _k

²

− K θ

˙

K θ

X

k∈R

( ˙ θ k − ω o )

²

≤ 0 (11)

In result, the potential function V ^M,N (x) defined in (10) is a Lyapunov function and the limit cycle (9) is stable. Let S be a set of all points in the system domain where ˙ V ^M,N (x) = 0, i.e.,

S = n

x ∈ D ¯

¯ r ˙ k = 0 for ∀k, θ ˙ k = ω

0

for ∀k / ∈ P o

(12) A point in an invariant set in S first needs to satisfy ¨ r k = 0 for ∀k, which implies r k = r

0

for ∀k (and hence the sets P and Q are null). At the same time, it also needs to satisfy ¨ θ k = 0 for ∀k, which implies the critical set of the (M, N)-potential function U ^M,N (θ). Therefore, the largest invariant set in S can be given by

M =

½ x ∈ S ¯

¯ r k = r

0

, ∂U ^M,N (θ)

∂θ k = 0 for ∀k

¾

From LaSalle’s invariant set theorem

¹⁵

, every trajectory starting in D approaches to the invariant set M.

In other words,

r k → r

0

, r ˙ k → 0, θ ˙ k → ω

0

for ∀ k ∈ {1, 2, ..., N}

and the phase configuration θ converges to one in the critical set of U ^M,N (θ) as t → ∞.

Now we prove an asymptotical stability of the splay and synchronized state by using Lyapunov indirect method. Let δx be a small disturbance from the state in the limit cycle (9)) denoted by

x

^∗

=



  r

0

1

0 θ

^∗

ω

0

1



  where 1 =



 1 .. . 1



 , 0 =



 0 .. . 0



 , θ

^∗

=



  θ

^∗₁

.. . θ

^∗

_N



 

Then its dynamics can be linearized as below.

δ x ˙ = F(θ

^∗

)δx =



 



O I O O

−K r I −K r

˙

I O O

O O O I

O O − ^K _r

2^θ

0

A(θ

^∗

) − ^K _r

2^θ˙ 0

I



 

 δx (13)

where (i, j)-element of the matrix A(θ

^∗

) in (13) is defined by A ij (θ

^∗

) = ∂

²

U ^M,N (θ)

∂θ i ∂θ j

¯ ¯

¯ θ

=

θ

^∗

= β _ij

^∗

− α

^∗

_i δ ij , i = 1, 2, . . . , N j = 1, 2, . . . , N where δ ij is the Kronecker delta and

α

^∗

_i = 1 N

X M

m=1

K m

X N

k=1

cos m(θ _k

^∗

− θ

^∗

_i ), β _ij

^∗

= 1 N

X M

m=1

K m cos m(θ

^∗

_j − θ _i

^∗

)

Note that, since β

^∗

_ij = β _ji

^∗

, the matrix A(θ

^∗

) is symmetric and so has N real eigenvalues denoted by λ _Aj (j = 1, 2, . . . , N ). The matrix F (θ

^∗

) has eigenvalues at

λ r± = −K r

˙

± p

K _r

²_˙

− 4K r

2 (14)

with multiplicity N associated with the relative distance dynamics, and also at

λ θ

j±

= −K θ

˙

± q

K _θ

²_˙

− 4r

₀²

K θ λ _Aj

2r

²₀

, j = 1, 2, . . . , N (15)

with multiplicity 1 associated with the phase dynamics. The eigenvalues λ r± have a strictly negative real part for any K r > 0 and K r

˙

> 0. Recall that λ _Aj is real. A real part of λ θ

j−

is strictly negative for any K θ > 0 and K θ

˙

> 0. A real part of λ θ

j+

is strictly positive for λ _Aj < 0, zero for λ _Aj = 0 and strictly negative for λ _Aj > 0. Consider the cases of θ

^∗

being the splay and the synchronized state.

i) Splay state (M = N ) : Without loss of generality, the splay phase distribution can be represented as θ

^∗

_k = 2πk/N for k = 1, 2, . . . , N . With this θ

^∗

, the matrix A(θ

^∗

) becomes a cyclic matrix given by

A(θ

^∗

) =



 

 

 

c

0

c

1

· · · c N

−2

c N

−1

c N−1 c

0

c

1

c N

−2

.. . c N−1 c

0

. .. .. .

c

2

. .. ... c

1

c

1

c

2

· · · c N

−1

c

0



 

 

 

where

c

0

= 1 N

X N

m=1

K m − K N , c i = 1 N

X N

m=1

K m cos mi 2π

N , i = 1, 2, . . . , N − 1

This cyclic matrix has the following eigenvalues and eigenvectors

¹⁶

. λ _Aj =

N−1 X

k=0

c k e ^ijk

^2πN

(16)

v _Aj = [ 1 e ^ij

^2πN

e ^i2j

^2πN

. . . e ^i(N

^−1)j2πN

] ^T (17) for j = 1, 2, . . . , N . By substituting c k into (16), λ _Aj is expanded as follows.

λ _Aj = c

0

+ 1 N

N−1 X

k=1

X N

m=1

K m cos mk 2π

N cos jk 2π N

= c

₀

+ 1 2N

N X

−1

k=1

X N

m=1

K _m cos (m + j)k 2π N + 1

2N

N−1 X

k=1

X N

m=1

K _m cos (m − j)k 2π N

= c

0

+ 1 2N

X N

l=1

( ¯ K l + ˜ K l ) Ã _N

X

k=1

cos lk 2π N − 1

!

where

K ¯ l =

½ N + l − j, 1 ≤ l ≤ j

l − j, j + 1 ≤ l ≤ N , K ˜ l =

½ −l + j, 1 ≤ l ≤ j − 1 N − l + j, j ≤ l ≤ N As we have

X N

k=1

cos lk 2π N =

½ 0, 1 ≤ l ≤ N − 1 N, l = N

the eigenvalues λ _Aj becomes λ _Aj =

½

₁

2

(K j + K N

−j

− 2K N ) > 0, j = 1, . . . , N − 1

0, j = N (18)

ii) Synchronized state (M = 1) : For the synchronized phase distribution, θ

^∗

_i = θ _j

^∗

for ∀i 6= j. Hence, the matrix A(θ

^∗

) becomes

A(θ

^∗

) = K

1

µ 1

N 11 ^T − I

¶

By solving the characteristic equation, an eigenvalue at 0 with an eigenvector v A = 1, and (N − 1) eigenvalues at −K

₁

> 0 with eigenvectors v _A satisfying 1 ^T v _A = 0 are obtained.

From the discussion above, in the both cases of the splay and synchronized states, the matrix A(θ

^∗

) has one zero eigenvalue and the other (N − 1) eigenvalues strictly positive. Also in the both cases, the eigenvector associated with the zero eigenvalue is v A = 1. This corresponds to a steady rotation within the same limit cycle (9). All the other positive eigenvalues of A(θ

^∗

) lead the eigenvalues λ θ

j±

in (15) to have a negative real part. From the Lyapunov indirect method, the limit cycle (9) of the splay and synchronized states is locally asymptotically stable when using the proposed controller (6-8) with M = N and M = 1 respectively.

C. Coordination Control Design with Virtual Leader at Center of Mass

In Section II.B, a coordination controller is designed with a virtual leader trajectory determined indepen- dently from the agents’ motion. However, in formation control problem, it is common to take a virtual leader at a center of mass of the team of agents. In this case, the VL position and velocity are given by

X vl = 1 N

X N

j=1

X j , X ˙ vl = 1 N

X N

j=1

X ˙ j (19)

This section modifies the coordination controller in Theorem II.3 by replacing a VL acceleration ¨ X vl by its

desired one, and analysis stability of the limit cycle (9) with the modified controller.

Theorem II.4 Define virtual leader position and velocity by (19), and suppose a controller X ¨ vl = u d gives a desired VL motion. With the control law (20) for each agent, a circle orbit around the VL with a radius r o , an angular velocity ω o and a phase configuration in a critical set of U ^M,N (θ) is stable. Furthermore, the splay state is locally asymptotically stable when M = N.

X ¨ k = u _rk

· cos θ k

sin θ k

¸ + u _θk

· − sin θ k

cos θ k

¸

+ u d (20)

where u _rk and u _θk given in (7-8).

Proof Consider a domain D

⁰

= {x ∈ D | r k > 0 ∀ k}. In this domain, with the controller (20), the relative distance and phase dynamics of the k-th agent become

¨

r k = −K r (r k − r o ) − K r

˙

r ˙ k − K rθ r k ( ˙ θ k − ω o )

²

+

· cos θ k

sin θ _k

¸ _T ³

u d − X ¨ vl

´

θ ¨ k = − 1 r _k

²

µ

K θ ∂U ^M,N (θ)

∂θ k + K θ

˙

( ˙ θ k − ω o )

¶

− (1 − K rθ ) r ˙ k

r k ( ˙ θ k − ω o ) + 1 r k

· − sin θ k

cos θ k

¸ _T ³

u d − X ¨ vl

´

Since u d − X ¨ vl = 0 when u _rk = u _θk = 0 for ∀ k, this system also has a limit cycle described in (9) which is included in the domain D

⁰

. On this limit cycle, the VL motion follows the dynamics of ¨ X vl = u d . Define a potential function as in (10), which has its unique minimum at the limit cycle (9). By using the relative distance and phase dynamics above, its time derivative is derived as follows.

V ˙ ^M,N (x) = − K r

˙

K θ

X N

k=1

˙ r

²

_k − K θ

˙

K θ

X N

k=1

( ˙ θ k − ω o )

²

+ 1 K θ

à _N X

k=1

˙ r k − ω o

X N

k=1

r k

· − sin θ k

cos θ k

¸! ^T

(u d − X ¨ vl ) (21)

From the definitions (4-5) and (19), the last term in (21) vanishes. Hence, V ˙ ^M,N (x) = − K r

˙

K θ

X N

k=1

˙ r _k

²

− K θ

˙

K θ

X N

k=1

r _k

²

( ˙ θ k − ω o )

²

≤ 0 (22) and the limit cycle (9) is stable.

Similarly to the previous section, an asymptotical stability of the splay state (when M = N ) is proven by using Lyapunov indirect method. Let δx be a small disturbance from the state in the limit cycle (9) with the splay phase distribution. Then its dynamics can be linearized as below.

δ x ˙ = ˜ F(θ

^∗

)δx =



 



O I O O

−K r I + C r −K r

˙

I + C r

˙

C θ C θ

˙

O O O I

D r D r

˙

− ^K _r

2^θ

0

A(θ

^∗

) + D θ − ^K _r

2^θ˙ 0

I + D θ

˙



 

 δx (23)

Define N × N matrices C and S whose (i, j)-elements are C ij = 1

N cos µ

(j − i) 2π N

¶

, S ij = 1 N sin

µ

(j − i) 2π N

¶

, i = 1, 2, . . . , N j = 1, 2, . . . , N Then, the matrices C

∗

and D

∗

(∗ = r, r, θ, ˙ θ) in ˜ ˙ F (θ

^∗

) are given by

C r = (K r + ω

²₀

)C, D r = 1

r

0

(K r + ω

₀²

)S C r

˙

= K r

˙

C + 2ω

0

S, D r

˙

= 1

r

0

(K r

˙

S − 2ω

0

C) C θ = −r

0

µ K θ

r

₀²

λ A1 + ω

²₀

¶

S, D θ = µ K θ

r

²₀

λ A1 + ω

₀²

¶ C C θ

˙

= 2r

0

ω

0

C − K θ

˙

r

0

S, D θ

˙

= 1 r

0

µ

2r

0

ω

0

S + K θ

˙

r

0

C

¶

where λ A1 =

¹₂

(K

1

+ K N

−1

− 2K N ) > 0 given in (18). Let λ be an eigenvalue of the matrix ˜ F (θ

^∗

) and v be its corresponding eigenvector. Then, λ and v can be obtained by solving the following equations.

(λ

²

+ K r

˙

λ + K r )v r = (λC r

˙

+ C r )v r + (λC θ

˙

+ C θ )v θ (24) (λ

²

+ K θ

˙

r

₀²

λ)v θ + K θ

r

²₀

A(θ

^∗

)v θ = (λD r

˙

+ D r )v r + (λD θ

˙

+ D θ )v θ (25) where v = [ v ^T _r λv ^T _r v ^T _θ λv ^T _θ ] ^T . When N = M = 2, we have

C = 1 2

· 1 −1

−1 1

¸

, S = O, A(θ

^∗

) = λ A1 C

By solving the equations (24, 25) with those matrices, the eigenvalues of the matrix ˜ F (θ

^∗

) are obtained as λ = 0, − K θ

˙

/r

₀²

, λ r± with multiplicity 1, and λ = ±iω

0

with multiplicity 2. When N = M > 2, the matrices C and S have the following characteristics;

CS = SC = 1

2 S, C

²

= −S

²

= 1

2 C (26)

which leads the relationship D

∗

= 2SC

∗

/r

0

. By using this relationship in the equations (24, 25), the eigenvalues are derived by λ = 0, − K θ

˙

/r

₀²

, λ θ

j±

for j = 2, 3, . . . , N − 2 with multiplicity 1, λ = λ r± with multiplicity N − 2, and λ = ±iω

0

, λ rθ± with multiplicity 2, where

λ rθ± = −(K r

˙

+ ^K _r

2^θ˙ 0

) ±

q

(K r

˙

+ ^K _r

2^θ˙

0

)

²

− 8(K r + ^K _r

2^θ 0

λ A

1

)

4 (27)

In any case, all the eigenvalues of the matrix ˜ F (θ

^∗

) have a strictly negative real part except λ = 0 and λ = ±iω

0

. While the zero eigenvalue corresponds to a steady rotation within the same limit cycle, the eigenvalues λ = ±iω

₀

correspond to a translational shift of the limit cycle which was not appeared in the previous case because the VL motion was given independently from the state x. From Lyapunov indirect method, the limit cycle (9) of the splay state is proven to be locally asymptotically stable with the proposed controller (20) with M = N.

D. Phase Constraint

In the previous two subsections, we have seen that the controllers (6, 20) stabilize a team of multi-agents on a circle orbit around a VL position with a radius r

0

and an angular velocity ω

0

, and that the splay and synchronized phase distributions are asymptotically stable. Whereas a relative phase configuration among all the agents is controlled to be a desired one with those controllers, an absolute phase trajectory cannot be specified. For example, in a case of ω

₀

= 0 and M = N , the agents will be equally distributed on a circle orbit around a virtual leader, but an orientation of the configuration is arbitrary. In this subsection, we impose a phase constraint on one of the agents in the team so that its phase trajectory tracks a given reference.

First, consider the case in which the virtual leader motion is known.

Corollary II.5 Let θ d be a phase reference where θ ˙ d = ω

0

. The following controller (28) stabilizes the limit cycle (9), with a phase trajectory of the j-th agent converging to θ d .

X ¨ k = u _rk

· cos θ k

sin θ k

¸ +

³

u _θk − δ jk

r j K θ

d

sin (θ j − θ d )

´ · − sin θ k

cos θ k

¸

+ ¨ X vl (28)

where u _rk and u _θk are given by (7, 8). Furthermore, the splay state is locally asymptotically stable when M = N .

Proof Define a new potential function by

W ^M,N (x) = V ^M,N (x) + K θ

_d

K θ

(1 − cos (θ _j − θ _d )) (29)

where V ^M,N (x) is given by (10). W ^M,N (x) is lower-bounded by K M N/2M

²

, which is attained uniquely at the limit cycle (9) with θ j = θ d . A time derivative of W ^M,N (x) can be derived as follows.

W ˙ ^M,N (x) = − K r

˙

K _θ X N

k=1

˙ r _k

²

− K θ

˙

K _θ X N

k=1

r _k

²

( ˙ θ k − ω o )

²

≤ 0 Therefore, W ^M,N (x) is a Lyapunov function and the limit cycle with θ _j = θ _d is stable.

An asymptotical stability of the splay state can be also proven. Consider a small disturbance δx from the desired splay state with the phase constraint θ j = θ d . Its dynamics can be linearized as below.

δ x ˙ = G(θ

^∗

)δx =



 



O I O O

−K r I −K r

˙

I O O

O O O I

O O − ^K _r

2^θ

0

A(θ

^∗

) − ^K _r

^θd2

0

e j e ^T _j − ^K _r

2^θ˙ 0

I



 

 δx (30)

The eigenvalues of G(θ

^∗

) associated with the relative distance dynamics remains the same as those of F (θ

^∗

) given in (14). Those associated with the relative phase dynamics need to satisfy

(λ

²

+ K θ

˙

r

²₀

λ)v θ + K θ

r

₀²

A(θ

^∗

)v θ + K θ

d

r

²₀

e j e ^T _j v θ = 0 (31) Without loss of generality, we can take j = 1 and θ d = 2π/N . Suppose v θ = P _N

k=1 α k v A

k

6= 0, where v A

k

given in (17). By using the fact of P _N

k=1 v A

k

/N = e

1

, the equation above becomes X N

k=1

µ

(λ

²

+ K θ

˙

r

₀²

λ + K θ

r

₀²

λ A

k

)α k + K θ

d

r

₀²

α ¯

¶

v A

k

= 0 (32)

where ¯ α = P _N

k=1 α _k /N and λ _A

_k

given by (18). This is satisfied when α _k for k = 1, 2, . . . , N is chosen so that every coefficients in (32) are zero. When ¯ α = 0, the following condition is required for any k = 1, 2, . . . , N .

α k = 0 or λ

²

+ K θ

˙

r

₀²

λ + K θ

r

₀²

λ A

k

= 0

This happens only when α N−i = −α i 6= 0 (i < N/2) and all the other α k ’s are zero. Such α’s give eigenvalues of λ = λ θ

i±

with multiplicity 1. Now suppose ¯ α 6= 0. By taking α N = 1 without loss of generality, we have

λ

²

+ K θ

˙

r

₀²

λ + K θ

d

r

₀²

α ¯ = 0 (33)

α k = K θ

d

α ¯ K θ

d

α ¯ − K θ λ A

k

= α ¯

¯

α − K _θ

⁰

λ A

k

, k = 1, 2, . . . , N − 1 (34) where K _θ

⁰

= K θ /K θ

d

> 0. By substituting this α k in the definition of ¯ α and by multiplying it by Q _N−1

i=1 (¯ α − K _θ

⁰

λ A

i

), ¯ α can be obtained by solving the following N th-order polynomial equation.

(N α ¯ − 1) Ã _N

₋₁

Y

i=1

(¯ α − K _θ

⁰

λ A

i

)

!

− α ¯

N X

−1

k=1





N Y

−1

i=1,i6=k

(¯ α − K _θ

⁰

λ A

i

)



 = N Ã

¯ α ^N +

N−1 X

i=0

(−1) ^N

⁻ⁱ

f i α ¯ ⁱ

!

= 0 (35)

where the coefficients f i > 0 for all i = 0, 1, . . . , N − 1 since K _θ

⁰

> 0 and λ A

k

> 0 for k = 1, 2, . . . , N − 1.

The equation (35) has (N + 1)/2 solutions for odd N and (N/2 + 1) solutions for even N since α k = α N−k , and they are all strictly positive. The eigenvalues of G(θ

^∗

) associated with each solution ¯ α can be obtained from (33), and they have a strictly negative real part since ¯ α > 0. Therefore, the splay state with θ j = θ d

is locally asymptotically stable. It should be noted that the zero eigenvalue disappeared because the phase constraint will be violated after the steady rotation.

Now we consider the case in which the VL position is given by a center of mass of the team of multi-agents.

Corollary II.6 VL position and velocity are defined by (19). The following controller (36) stabilizes the limit cycle (9) with θ j = θ d and the VL motion following X ¨ d = u d .

X ¨ k = u _rk

· cos θ k

sin θ k

¸ +

³

u _θk − δ jk

r j K θ

d

sin (θ j − θ d )

´ · − sin θ k

cos θ k

¸

+ u d (36)

where u _rk and u _θk are given by (7, 8). Furthermore, the splay state is locally asymptotically stable when M = N .

Proof Consider a domain D

⁰

= {x ∈ D | r k > 0 ∀ k} which includes the limit cycle. Define a potential function as in (29). Similarly to the proofs of Theorem II.4 and Corollary II.5, W ^M,N (x) is a Lyapunov function and the limit cycle with θ j = θ d is stable.

Consider the case M = N . Dynamics of the small disturbance from the desired splay state with the phase constraint θ j = θ d can be linearized as follows.

δ x ˙ = ¯ G(θ

^∗

)δx =



 



O I O O

−K r I + C r −K r

˙

I + C r

˙

C ¯ θ C θ

˙

O O O I

D _r D _r

_˙

− ^K _r

2^θ

0

A(θ

^∗

) − ^K _r

^θd2

0

e _j e ^T _j + ¯ D _θ − ^K _r

2^θ˙ 0

I + D θ

˙



 

 δx (37)

where C

∗

and D

∗

(∗ = r, r, θ, ˙ θ) are defined in Section II.C, and ˙ C ¯ θ = C θ − K θ

d

r

0

Se j e ^T _j , D ¯ θ = D θ + K θ

d

r

²₀

Ce j e ^T _j Without loss of generality, we take j = 1 and θ d = 2π/N.

When N = M = 2, the matrix ¯ G(θ

^∗

) has eigenvalues of λ = λ r± ,

λ θ

d±

= −K θ

˙

± q

K _θ

²_˙

− 2r

²₀

K θ

d

2r

²₀

with multiplicity 1 and λ = ±iω

0

with multiplicity 2. When N = M > 2, recall that we have relationships of (26) and D

∗

= 2SC

∗

/r

0

which holds for the new matrices ¯ C θ and ¯ D θ . The eigenvalues associated purely with the relative distance dynamics (i.e., v θ = 0) remain the same as those of the matrix ¯ F(θ

^∗

), which is λ = λ r± with multiplicity N − 2. The eigenvalues associated purely with the relative phase dynamics need to satisfy the equation (31) as well as (λC θ

˙

+ ¯ C θ )v θ = 0. The two equations give the eigenvalues of λ θ

i±

defined in (15) when v θ = v A

i

− v A

N−i

for 2 ≤ i < N/2. Now consider the eigenvalues associated with the coupled distance-phase dynamics. From the properties of C and S given in (26), we obtain the equation Cv r = v r /2 which leads to v r = α

1

v A

1

+ α N

−1

v A

N−1

6= 0. Suppose v θ = P _N

k=1 β k v A

k

, then λ, α

1

, α N

−1

and β k ’s can be derived by solving

¡ (λ

²

− ω

₀²

) − i2λω

0

¢ (α

1

− ir

0

β

1

) v A

1

+ ¡

(λ

²

− ω

₀²

) + i2λω

0

¢ (α N

−1

+ ir

0

β N

−1

) v A

N−1

= 0 (38) X N

k=1

³

(λ

²

+ K θ

˙

r

₀²

λ + K θ

r

²₀

λ A

k

)β k + K θ

d

r

₀²

β ¯ ´

v A

k

= i

r

0

(λ

²

+ K r

˙

λ + K r )(α

1

v A

1

− α N−1 v A

N−1

) (39) where ¯ β = P _N

k=1 β k /N. (38) is satisfied either i) when λ = iω

0

and β N

−1

= iα N

−1

/r

0

, ii) when λ = −iω

0

and β

1

= −iα

1

/r

0

, or iii) when β

1

= −iα

1

/r

0

and β N−1 = iα N

−1

/r

0

. (39) is satisfied if γ k = 0 for ∀ k, where

γ k = (λ

²

+ K θ

˙

r

₀²

λ + K θ

r

₀²

λ A

k

)β k + K θ

d

r

₀²

β ¯ +

 



− _r ⁱ

0

(λ

²

+ K r

˙

λ + K r )α

1

, k = 1 + _r ⁱ

0

(λ

²

+ K r

˙

λ + K r )α N

−1

, k = N − 1

0, otherwise

When ¯ β = 0, the two equations are satisfied only in the case of iii) with β

1

= −β N−1 and β k = 0 for k = 2, 3, . . . , N − 2, N . In this case, the eigenvalues are given by λ = λ rθ± in (27). Now suppose ¯ β 6= 0. In the case of i),

K θ

d

r

²₀

β ¯ = i r

0

µ

2ω

²₀

− K r − K θ

r

₀²

λ A

1

− iω

0

(K r

˙

+ K θ

˙

r

₀²

)

¶

α N

−1